Precalculus Examples

$\log_{27} (x) = - \frac{4}{3}$

Step 1

Rewrite $\log_{27} (x) = - \frac{4}{3}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$27^{- \frac{4}{3}} = x$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x = 27^{- \frac{4}{3}}$ .

$x = 27^{- \frac{4}{3}}$

Step 2.2

Simplify $27^{- \frac{4}{3}}$ .

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Step 2.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{27^{\frac{4}{3}}}$

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

Rewrite $27$ as $3^{3}$ .

$x = \frac{1}{{(3^{3})}^{\frac{4}{3}}}$

Step 2.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{1}{3^{3 (\frac{4}{3})}}$

Step 2.2.2.3

Cancel the common factor of $3$ .

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Step 2.2.2.3.1

Cancel the common factor.

$x = \frac{1}{3^{3 (\frac{4}{3})}}$

Step 2.2.2.3.2

Rewrite the expression.

$x = \frac{1}{3^{4}}$

Step 2.2.2.4

Raise $3$ to the power of $4$ .

$x = \frac{1}{81}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{81}$

Decimal Form:

$x = 0.01234567 \dots$